Not him, but he definitely could have had something with PB before they made it as awkward as it could possibly be, then introduced Flame Princess, then crashed that relationship with no survivors, then did Breezy, then had a period where Finn's entire character arc was learning why he's so bad at girls, and finally started teasing him and Huntress Wizard.
Basically, turning Finn's love life into a cyclone of misery ruined the show, and here's how:
The purpose of a narrative is to create a desire for fulfillment in the audience, and then to provide that fulfillment. Once you build towards something in your story, a conflict, a relationship, a mystery, a quest, or character development, even if it's just one hint or step in some direction, your audience will expect you to walk them to the end of that path.
So, we get this buildup in AT. Finn has a huge crush on Bubblegum. Okay cool, how will this be resolved? Ahhh, they're going to de-age her. No wait, they re-aged her. It seems like we're walking backwards on the path now. Uh, oh, she pretty firmly told him he's too young for her, but he's still in to her. I still need my resolution. Oh shoot, they're introducing a new love interest! I see, this path was taking me somewhere else entirely! Now this plot thread is resolved! Oh wait, Finn just torpedoed his relationship in a very weird way with no buildup. Looks like I still need resolution. Wait, does he like PB again? Which path am I on? No wait, he still likes Flame Princess, but she's hanging out with a pastry! I'm all over the place and there's no resolution in sight! Now Finn has PTSD and is smooching girls! There was no path, there never was a path! Looks like he is now learned to forget about girls and focus on his own deal. Definitely not a satisfying resolution, but I'll take it. Oh shoot, now he's being teased with a new love interest? WHERE'S THE END OF THIS PATH?
TD;DR Finn's romance arc is a narrative abomination.